The most common and most likely answer to this crossword clue is the 11 letter word EGO-SURFING
I've seen this crossword clue in Independent
E
G
O
-
S
U
R
F
I
N
G
Buddy explains!
The straight part
We know that "a vain exercise of modern times" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a vain exercise of modern times = ego-surfing.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EGO-SURFING mean?
EGO-SURFING noun- The act of using a search engine to find mentions of one's name on the Internet.